Average Error: 20.9 → 18.2
Time: 12.3s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -3.226296127080268 \cdot 10^{+302} \lor \neg \left(z \cdot t \leq 8.510999031196353 \cdot 10^{+290}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right) + \sin y \cdot \left(\sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)} \cdot \left(\sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)} \cdot \sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}
↓
\begin{array}{l}
\mathbf{if}\;z \cdot t \leq -3.226296127080268 \cdot 10^{+302} \lor \neg \left(z \cdot t \leq 8.510999031196353 \cdot 10^{+290}\right):\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right) + \sin y \cdot \left(\sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)} \cdot \left(\sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)} \cdot \sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\

\end{array}
(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* z t) -3.226296127080268e+302)
         (not (<= (* z t) 8.510999031196353e+290)))
   (- (* (* 2.0 (sqrt x)) (- 1.0 (* 0.5 (* y y)))) (/ a (* b 3.0)))
   (-
    (*
     (* 2.0 (sqrt x))
     (+
      (*
       (cos y)
       (cos
        (*
         (* (cbrt (/ (* z t) 3.0)) (cbrt (/ (* z t) 3.0)))
         (/ (cbrt (* z t)) (cbrt 3.0)))))
      (*
       (sin y)
       (*
        (cbrt
         (sin
          (*
           (* (cbrt (/ (* z t) 3.0)) (cbrt (/ (* z t) 3.0)))
           (/ (cbrt (* z t)) (cbrt 3.0)))))
        (*
         (cbrt
          (sin
           (*
            (* (cbrt (/ (* z t) 3.0)) (cbrt (/ (* z t) 3.0)))
            (/ (cbrt (* z t)) (cbrt 3.0)))))
         (cbrt
          (sin
           (*
            (* (cbrt (/ (* z t) 3.0)) (cbrt (/ (* z t) 3.0)))
            (/ (cbrt (* z t)) (cbrt 3.0))))))))))
    (/ a (* b 3.0)))))
double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos(y - ((z * t) / 3.0))) - (a / (b * 3.0));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z * t) <= -3.226296127080268e+302) || !((z * t) <= 8.510999031196353e+290)) {
		tmp = ((2.0 * sqrt(x)) * (1.0 - (0.5 * (y * y)))) - (a / (b * 3.0));
	} else {
		tmp = ((2.0 * sqrt(x)) * ((cos(y) * cos((cbrt((z * t) / 3.0) * cbrt((z * t) / 3.0)) * (cbrt(z * t) / cbrt(3.0)))) + (sin(y) * (cbrt(sin((cbrt((z * t) / 3.0) * cbrt((z * t) / 3.0)) * (cbrt(z * t) / cbrt(3.0)))) * (cbrt(sin((cbrt((z * t) / 3.0) * cbrt((z * t) / 3.0)) * (cbrt(z * t) / cbrt(3.0)))) * cbrt(sin((cbrt((z * t) / 3.0) * cbrt((z * t) / 3.0)) * (cbrt(z * t) / cbrt(3.0))))))))) - (a / (b * 3.0));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.9
Target18.9
Herbie18.2
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 z t) < -3.226296127080268e302 or 8.5109990311963532e290 < (*.f64 z t)

    1. Initial program 62.1

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]

    2. Taylor expanded around 0 45.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - 0.5 \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]

    3. Simplified45.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - 0.5 \cdot \left(y \cdot y\right)\right)} - \frac{a}{b \cdot 3}\]

    if -3.226296127080268e302 < (*.f64 z t) < 8.5109990311963532e290

    1. Initial program 14.7

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt_binary6414.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}}\right) - \frac{a}{b \cdot 3}\]
    4. Using strategy rm

    5. Applied cbrt-div_binary6414.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \color{blue}{\frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}}\right) - \frac{a}{b \cdot 3}\]
    6. Using strategy rm

    7. Applied cos-diff_binary6414.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right) + \sin y \cdot \sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)\right)} - \frac{a}{b \cdot 3}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt_binary6414.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right) + \sin y \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)} \cdot \sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)}\right) \cdot \sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)}\right)}\right) - \frac{a}{b \cdot 3}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -3.226296127080268 \cdot 10^{+302} \lor \neg \left(z \cdot t \leq 8.510999031196353 \cdot 10^{+290}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right) + \sin y \cdot \left(\sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)} \cdot \left(\sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)} \cdot \sqrt[3]{\sin \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \frac{\sqrt[3]{z \cdot t}}{\sqrt[3]{3}}\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))